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\begin{document}
\begin{frame}
The region $R$ is bounded by the curves
$$ y=x^3 \hskip 20pt y=8 \hskip 20pt x=0 $$
Sketch $R$. For the following rotational axes, {\bf set-up} two integrals for
the volume of the solid generated by revolving $R$ about the indicated axis,
one representing the washer method and one the cylindrical shells method.
\begin{itemize}
\item[\bf (a)] $x$-axis.
\item[\bf (b)] $y$-axis.
\item[\bf (c)] $y=5$.
\item[\bf (d)] $x=-2$.
\end{itemize}
\end{frame}
\begin{frame}
The region $R$ is bounded by the curves
$$ y=1+\sin(x) \hskip 20pt y=1 \hskip 20pt x=0 \hskip 20pt x=2 $$
Sketch $R$. For the following rotational axes, {\bf set-up} two integrals for the
volume of the solid generated by revolving $R$ about the indicated axis, one
representing the washer method and one the cylindrical shells method.
\begin{itemize}
\item[\bf (a)] $x$-axis.
\item[\bf (b)] $y$-axis.
\item[\bf (c)] $y=-1$.
\end{itemize}
\end{frame}
\begin{frame}
The triangular region with vertices $(0,2)$, $(1,0)$, and $(0,1)$ is rotated about
the line $x=4$. Find the volume of the solid generated by this rotation.
\end{frame}
\begin{frame}
Let $B$ be the region bounded by the graphs of $x=y^2$ and $x=9$. Sketch $B$. For each part
below, find the volume of the solid that has $B$ as its base if every cross section by a
plane perpendicular to the $x$-axis is
\begin{itemize}
\item[\bf (a)] a square.
\item[\bf (b)] a semicircle with diameter lying on $B$.
\item[\bf (c)] an equilateral triangle.
\end{itemize}
\end{frame}
\begin{frame}
Find the volume of a wedge cut out of a cylinder of radius $r$ if the angle between
the top and bottom of the wedge is $\frac{\pi}{6}$.
\end{frame}
\end{document}